Download Combinatorial Circuits, Boolean Algebras

CHAPTER 8
Boolean Algebras
8.1. Combinatorial Circuits
8.1.1. Introduction. At their lowest level digital computers handle only binary signals, represented with the symbols 0 and 1. The
most elementary circuits that combine those signals are called gates.
Figure 8.1 shows three gates: OR, AND and NOT.
x1
OR GATE
x1
x2
x1
x2
x2
x1
AND GATE
x2
NOT GATE
x
x
Figure 8.1. Gates.
Their outputs can be expressed as a function of their inputs by the
following logic tables:
x1 x2 x1 ∨ x2
1 1
1
1 0
1
0 1
1
0 0
0
OR GATE
118
8.1. COMBINATORIAL CIRCUITS
119
x1 x2 x1 ∧ x2
1 1
1
1 0
0
0 1
0
0 0
0
AND GATE
x
1
0
x̄
0
1
NOT GATE
These are examples of combinatorial circuits. A combinatorial circuit is a circuit whose output is uniquely defined by its inputs. They
do not have memory, previous inputs do not affect their outputs. Some
combinations of gates can be used to make more complicated combinatorial circuits. For instance figure 8.2 is combinatorial circuit with
the logic table shown below, representing the values of the Boolean
expression y = (x1 ∨ x2 ) ∧ x3 .
x1
y
x2
x3
Figure 8.2. A combinatorial circuit.
x1 x2 x3 y = (x1 ∨ x2 ) ∧ x3
1 1 1
0
1 1 0
1
1 0 1
0
1 0 0
1
0 1 1
0
0 1 0
1
0 0 1
1
0 0 0
1
However the circuit in figure 8.3 is not a combinatorial circuit. If
x1 = 1 and x2 = 0 then y can be 0 or 1. Assume that at a given time
y = 0. If we input a signal x2 = 1, the output becomes y = 1, and
8.1. COMBINATORIAL CIRCUITS
120
stays so even after x2 goes back to its original value 0. That way we
can store a bit. We can “delete” it by switching input x1 to 0.
x1
y
x2
Figure 8.3. Not a combinatorial circuit.
8.1.2. Properties of Combinatorial Circuits. Here Z2 = {0, 1}
represents the set of signals handled by combinatorial circuits, and the
operations performed on those signals by AND, OR and NOT gates are
represented by the symbols ∧, ∨ and ¯ respectively. Then their properties are the following (a, b, c are elements of Z2 , i.e., each represents
either 0 or 1):
1. Associative
(a ∨ b) ∨ c = a ∨ (b ∨ c)
(a ∧ b) ∨ c = a ∧ (b ∧ c)
2. Commutative
a∨b=b∨a
a∧b=b∧a
3. Distributive
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
4. Identity
5. Complement
a∨0=a
a∧1=a
a ∨ ā = 1
a ∧ ā = 0
A system satisfying those properties is called a Boolean algebra.
Two Boolean expressions are defined to be equal is they have the
same values for all possible assignments of values to their literals. Example: x ∨ y = x̄ ∧ ȳ, as shown in the following table:
8.1. COMBINATORIAL CIRCUITS
x
1
1
0
0
121
y x ∨ y x̄ ∧ ȳ
1
0
0
0
0
0
1
0
0
0
1
1
8.1.3. Boolean Algebras. Here we deal with general Boolean
algebras; combinatorial circuits are an example, but there are others.
A Boolean algebra B = (S, +, ·, 0 , 0, 1) is a set S containing two
distinguished elements 0 and 1, two binary operators + and · on S,
and a unary operator 0 on S, satisfying the following properties (x, y,
z are elements of S):
1. Associative
(x + y) + z = x + (y + z)
(x · y) + z = x · (y · z)
2. Commutative
x+y =y+x
x·y =y·x
3. Distributive
x · (y + z) = (x · y) + (x · z)
x + (y · z) = (x + y) · (x + z)
4. Identity
x+0=x
x·1=x
5. Complement
x + x0 = 1
x · x0 = 0
Example: (Z2 , ∨, ∧, ¯, 0, 1) is a Boolean algebra.
Example: If U is a universal set and P(U )= the power set of S
(collection of subsets of S) then (P(U ), ∪, ∩, ¯, ∅, U ). is a Boolean
algebra.
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8.1.4. Other Properties of Boolean Algebras. The properties
mentioned above define a Boolean algebra, but Boolean algebras also
have other properties:
1. Idempotent
2. Bound
3. Absorption
x+x=x
x·x=x
x+1=1
x·0=0
x + xy = x
x · (x + y) = x
4. Involution
5. 0 and 1
(x0 )0 = x
00 = 1
10 = 0
6. De Morgan’s
(x + y)0 = x0 · y 0
(x · y)0 = x0 + y 0
For instance the first idempotent law can be proved like this: x =
x + 0 = x + x · x0 = (x + x) · (x + x0 ) = (x + x) · 1 = x + x.